Question

Newton's law of gravitation states that any two bodies attract each other with a force that is inversely proportional to the square of the distance between them. The force of attraction between the earth and some object is called the weight of that object. The law of gravitation states, then, that the weight of an object is inversely proportional to the square of its distance from the center of the earth. If a person weighs 150 lb on the surface of the earth (assume this to be 3960 mi from the center), how much will he weigh 1500 mi above the surface of the earth?

Short answer

The person would weigh approximately 78.947 lb, 1500 mi above the surface of the earth.

Step-by-step solution

Understand the Law of Gravitation

Newton's Law of Gravitation indicates that the force of attraction (weight) between two objects is inversely proportional to the square of the distance (r) between them. Mathematically, this can be expressed as F ∝ 1/r^2.

Calculate the Distance from the Center of the Earth at the New Position

To find the new weight at 1500 mi above the Earth's surface, add the distance from the Earth's surface to the center (3960 mi) to the additional height (1500 mi). The total distance r from the center is thus 3960 + 1500 = 5460 mi.

Set Up the Proportionality for the Weights

Since weight is proportional to 1/r^2, we can set up a proportion using the weight on the surface W1 and the weight at the new distance W2: W1/W2 = (r2/r1)^2, where r1 is the original distance, and r2 is the new distance from the center.

Calculate the New Weight

Using the weights and distances: 150 lb / W2 = (5460 mi / 3960 mi)^2. Solve for W2 to find the new weight 1500 mi above the Earth's surface. W2 = 150 lb * (3960 mi / 5460 mi)^2.

Solve for the Unknown Weight

Calculate W2: W2 = 150 lb * (3960^2 / 5460^2) = 150 lb * (15681600 / 29811600) = 150 lb * (0.52631579) ≈ 78.947 lb. The person would weigh approximately 78.947 lb, 1500 mi above the surface of the earth.

Key concepts

Gravitational Force

Gravitational force is a fundamental interaction that occurs between any two objects with mass. According to Sir Isaac Newton's Law of Gravitation, every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This force is what we commonly refer to as gravity.

In more concrete terms, this is why objects fall to the ground when dropped and why planets orbit around stars. For example, the gravitational pull of the Earth is what keeps the moon in orbit and causes the phenomenon of tides. Moreover, this force is what gives us weight, or the force of gravity on an object at the Earth's surface.

Inverse Square Law

The inverse square law is a physical principle stating that a specified physical quantity or strength is inversely proportional to the square of the distance from the source of that physical quantity. This law applies to various forms of radiation and forces, including the gravitational force mentioned in Newton's Law of Gravitation.

The importance of this law is best illustrated with an example: if you double the distance between two objects, the gravitational force between them decreases by a factor of four (2 squared). This relationship is crucial in understanding the effect of distance on gravitational forces and can be applied to other forces, such as electromagnetic forces, which also follow an inverse square law.

Weight Calculation

Weight calculation in the context of gravity refers to determining the gravitational force exerted on an object. Since weight is the force due to gravity, it can be calculated using Newton's Law of Gravitation. For an object on or near the surface of Earth, the weight is simply the mass of the object multiplied by the acceleration due to gravity (approximately 9.81 m/s^2 on Earth).

However, as the distance from Earth increases, this weight decreases following the inverse square law. To calculate the weight at a different distance from Earth, we need to compare the square of the ratios of the distances from the center of the Earth and multiply that by the original weight. This method allows for the calculation of an object's weight at various altitudes above the Earth's surface.

Proportional Relationship

A proportional relationship denotes a direct or inverse correlation between two quantities. In the context of gravitational forces, the weight of an object is inversely proportional to the square of its distance from the center of the Earth. This means that the closer you are to Earth's center, the stronger the gravitational pull and, thus, the heavier you would weight.

This relationship is central to solving problems involving changing distances from a mass, such as Earth. By setting up a proportionality equation, one can easily solve for unknown variables. Mathematically, if two variables are proportional, they can be expressed as a constant ratio, which remains the same regardless of the specific values of those variables. This foundational concept allows one to calculate changes in gravitational force and hence weight, as illustrated in the given exercise where the weight of a person is determined at different distances from the center of the Earth.